In this paper an implicit characterization of the complexity classes k-EXP and k-FEXP, for k>=0, is given, by a type assignment system for a stratified lambda-calculus, where types for programs are witnesses of the corresponding complexity class. Types are formulae of Elementary Linear Logic (ELL), and the hierarchy of complexity classes k-EXP is characterized by a hierarchy of types. (This on-line version contains an appendix of 14 pages with full proofs of the statements)